The Cozy Car Company created a new model of car especially for Japan and Vietnam, which they start shipping to both countries at the same time. The company had shipped $2$ of that model of car to Japan by the end of the first month, and the cumulative total of cars they've shipped triples each month. The company had shipped $5$ of that model of car to Vietnam by the end of the first month, and the cumulative total of cars they've shipped increases by $11$ cars each month. The company's board assembles at the end of each month to review the cumulative total number of cars shipped to each country. What is the first month in which the board will see the cumulative total number of cars shipped to Japan exceeded the cumulative total in Vietnam?
Solution: Notice that Japan's cumulative total grows exponentially while Vietnam's cumulative total grows linearly. This means Japan's cumulative total is bound to exceed Vietnam's cumulative total at some point. Let's start calculating the cumulative total number of cars of the new model shipped to each country to see when that happens. Month Japan Vietnam (Multiply by $3$ each month.) (Add $11$ each month.) $1$ $2$ $5$ $2$ $6$ $16$ $3$ $18$ $27$ $4$ $54$ $38$ In conclusion, the cumulative number of cars of that model shipped to Japan will first exceed the cumulative total shipped to Vietnam in month number $4$.